Global Positioning System Reference
In-Depth Information
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the general linear hypothesis test. For the observation equation model with additional
conditions between the parameters, one has
v 1 =
A 1 x
+ 1
(4.277)
H 0 : A 2 x
+ 2 =
o
(4.278)
Equation (4.278) expresses the zero hypothesis H 0 . The solution of the combined
adjustment is found in Table 4.5. Adjusting (4.277) alone results in v T Pv , which
has a chi-square distribution with n
r degrees of freedom according to (4.273). The
v T Pv resulting from the condition (4.278) is
change
= A 2 x + 2 T
T A 2 x + 2
v T Pv
(4.279)
[13
The expression in (4.279) differs from (4.269) in two respects. First, the matrix T dif-
fers; i.e., the matrix T in (4.279) does not contain the P 2 matrix. Second, the quantity
2 is not a random variable. These differences, however, do not matter in the proof
of stochastic independence of v T Pv and
Lin
0.6
——
Sho
PgE
v T Pv . Analogously to (4.269), we can
v T Pv in (4.279) as a function of a new random variable z 3 . The
proof for stochastic independence follows the same lines of thought as given before
(for the case of additional observations). Thus, just as (4.274) is the basis for testing
two groups of observations, the basic test for the general linear hypothesis (4.278) is
express the change
v T Pv (n 1
r)
F n 2 ,n 1 r
(4.280)
v T Pv
n 2
[13
v T Pv implies that the null hypothesis (4.278) is acceptable; i.e., the con-
ditions are in agreement with the observations. The conditions do not impose any
distortions on the adjustment. The rejection criterion is based on the one-tail test at
the upper end of the distribution. Thus, reject H 0 at a 100
A small
α
% significance level if
F>F n 2 ,n 1 r, α
(4.281)
The general formulation of the null hypothesis in (4.278) makes it possible to
test any hypothesis on the parameters, so long as the hypothesis can be expressed
in a mathematical equation. Nonlinear hypotheses must first be linearized. Simple
hypotheses could be used to test whether an individual parameter has a certain nu-
merical value, whether two parameters are equal, whether the distance between two
stations has a certain length, whether an angle has a certain size, etc. For example,
consider the hypothesis
H 0 : x
x T
=
o
(4.282)
H 1 : x
x T
=
o
(4.283)
 
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